Depth and Associated Primes of Modules over a Ring

Abstract

This thesis consists of three main topics. In the first topic, we let $R$ be a commutative Noetherian ring, $I,J$ ideals of $R$, $M$ a finitely generated $R$-module and $F$ an $R$-linear covariant functor. We ask whether the sets $\operatorname{Ass}_R F(M/I^n M)$ and the values $\operatorname{depth}_J F(M/I^n M)$ become independent of $n$ for large $n$. In the second topic, we consider rings of the form $R = k[x^a,x^{p_1}y^{q_1}, \ldots,x^{p_t}y^{q_t},y^b]$, where $k$ is a field and $x,y$ are indeterminates over $k$. We will try to formulate simple criteria to determine whether or not $R$ is Cohen-Macaulay. Finally, in the third topic we introduce and study basic properties of two types of modules over a commutative Noetherian ring $R$ of positive prime characteristic. The first is the category of modules of finite $F$-type. They include reflexive ideals representing torsion elements in the divisor class group. The second class is what we call $F$-abundant modules. These include, for example, the ring $R$ itself and the canonical module when $R$ has positive splitting dimension. We prove many facts about these two categories and how they are related. Our methods allow us to extend previous results by Patakfalvi-Schwede, Yao and Watanabe. They also afford a deeper understanding of these objects, including complete classifications in many cases of interest, such as complete intersections and invariant subrings.